Asymptotic uniformity conjecture for constellations under the gap-cycle recursion

Let ${\mathcal G}(p)$ denote the cycle of gaps at a sieve stage, let $s$ be a constellation in ${\mathcal G}(p)$, and let $\sigma(s)$ be the sum of its gaps. Assume $\sigma(s)<2p$. Asymptotic uniformity conjecture. Under the recursion on the cycle of gaps, all such constellations tend toward a uniform distribution in ${\mathcal G}(P)$ for all primes $P\gg p$. This is the strengthened later formulation of the paper's uniformity assumption; the recursion suggests it, but no proof of the asserted convergence is supplied.